I was forced to put my gun safe in the living room of our house. There is a basement underneath; however, there is ductwork built between the joists  underneath the safe. I'll reinforce the joists with vertical 4x4s. Since I can't access the subfloor from the basement due to the ductwork, is it worth putting a few large screws inside the safe to go through the hardwood floor and subfloor? I can't see it doing all that much. Any thoughts?

You'll want to bolt it to the floor.  A: It prevents it from getting carried away in a burg.  B: It prevents it from tipping over and causing injury.

If you can't access the screws to reinforce, my guess is rocking your safe would make quick tear out from the sub-floor, just my .02

Insurance and a alarm would be my choice.

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not really sure why you just didn't put the safe in the basement. that is where mine has been for the last 15 years

If you have room to get under the floor to reinforce the joists with 4x4 then I think you have some options. Is there anyway that you could align at least two of the bolt down holes with a long length of all thread? You could run lengths of all thread through the subfloor downward to wood blocking or steel u channel / unistrut run perpendicular to the floor joist. There is often a slight gap between the joists and the ducting. This could all be done with fairly simple tools like power drills, hack saws and files. You don't need complete access you just need a straight shot down at some point to run the rods to the u channel / unistrut. Does the ducting extend below the joists at all?

You should also consider lead bars for the bottom of the safe. Although too much weight will eventually require additional joists. Pictures of your set up would be very helpful for brainstorming.